It depends. If the host box does NAT or routing, then the answer is no. If the host box is merely a bridge, then yes, it's possible. You just need to create a rule watching traffic from the local host to any remote address via the Internet NIC.

DHCP and NAT are independent things. Try to create a rule watching traffic from local host to any IP address on the external NIC and see whether it registers all traffic or the host box's traffic only.